Spatial Vision, Vol. 19, No. 5, pp. 439– 457 (2006)
 VSP 2006.
Also available online - www.brill.nl/sv
Perceptual distance and the constancy of size and
stereoscopic depth
LLOYD KAUFMAN 1,2,∗ , JAMES H. KAUFMAN 3 , RICHARD NOBLE 4 ,
STEFAN EDLUND 3 , SUNHEE BAI 1 and TERESA KING 1
1 Department of Psychology, C. W. Post Campus, Long Island University, Brookville, NY, USA
2 Department of Psychology, New York University, New York, NY, USA
3 IBM Research, Almaden Research Center, IBM, San Jose, CA, USA
4 Department of Computer Science, C. W. Post Campus, Long Island University, Brookville, NY,
USA
Received 31 August 2005; accepted 23 January 2006
Abstract—The relationship between distance and size perception is unclear because of conflicting
results of tests investigating the size–distance invariance hypothesis (SDIH), according to which
perceived size is proportional to perceived distance. We propose that response bias with regard to
measures of perceived distance is at the root of the conflict.
Rather than employ the usual method of magnitude estimation, the bias-free two-alternative forced
choice (2AFC) method was used to determine the precision (1/σ ) of discriminating depth at different
distances. The results led us to define perceptual distance as a bias free power function of physical
distance, with an exponent of ∼0.5. Similar measures involving size differences among stimuli of
equal angular size yield the same power function of distance. In addition, size discrimination is
noisier than depth discrimination, suggesting that distance information is processed prior to angular
size.
Size constancy implies that the perceived size is proportional to perceptual distance. Moreover,
given a constant relative disparity, depth constancy implies that perceived depth is proportional to the
square of perceptual distance. However, the function relating the uncertainties of depth and of size
discrimination to distance is the same. Hence, depth and size constancy may be accounted for by the
same underlying law.
Keywords: Perceived distance; perceptual distance; size constancy; depth constancy; 2AFC; magnitude estimation; depth discrimination; size discrimination.
INTRODUCTION
In Holway and Boring’s (1941) classic study of size constancy subjects adjusted the
diameter of a nearby disc to match the diameter of a relatively distant disc. Their
∗ To
whom correspondence should be addressed. E-mail: lloyd.kaufman@liu.edu
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matches approximated the objective (linear) size of the more distant disc rather than
its angular size. However, as cues to distance were reduced, subjects increasingly
tended toward matching the angular sizes of the two discs. This result reinforced
the widespread assumption that the perception of objective size is possible because
the perceptual system takes distance into account.
For example, the size–distance invariance hypothesis (SDIH) describes one way
in which perceived objective size may depend upon perceived distance (cf. Epstein
et al., 1961; Kilpatrick and Ittelson, 1953; Sedgwick, 1986). SDIH holds that
some function of retinal size combines multiplicatively with perceived distance to
allow the perception of objective size, and, therefore, size constancy. In its simplest
approximation:
S = D tan α,
(1)
where S is the perceived size (in one dimension) of a surface oriented normal to the
line of sight; D is the perceived distance to that object; and α is the angular size of
the object.
SDIH is actually a restatement of how to compute the objective size of a distant
object from its angular size and its physical distance. That is, according to Euclid’s
law, the angular size of an object of constant linear size is inversely proportional
to its distance. Given the angular size of an object and its distance, it is a matter
of simple arithmetic to compute its distal size. The main difference between
this and SDIH is that perceived size and perceived distance replace physical size
and distance. Presumably, according to SDIH, the brain engages in a similar
computation.
The assumption that perceiving objective size requires taking perceived distance
into account begs a number of questions. For one, how do we define perceived
distance? In psychophysics such a definition describes a relationship between the
perceived magnitude of a stimulus and its physical magnitude.
As is well known, the method of magnitude estimation typically reveals that
sensory magnitudes are power functions of the physical magnitudes of sensory
stimuli (cf. Stevens, 1957). In many studies this method indicated that perceived
distance is a power function of physical distance, with an exponent ranging from
0.7 to about 1.5. (cf. Baird and Biersdorf, 1967; Baum and Jonides, 1979; Da
Silva and Fukusima, 1986; Higashiyama and Shimono, 1994; 2004; Teghtsoonian
and Teghtsoonian, 1969; Wagner, 1985). The precise value of the exponent varies
widely across subjects as well as experiments. Künnapas (1960) and Teghtsoonian
and Teghtsoonian (1969), in indoor experiments, found that estimates of distance are
represented by a power function of physical distance, with exponents ranging from
1.15 to 1.47. Teghtsoonian and Teghtsoonian (1970), working outdoors, obtained
exponents that ranged from 0.85 to 0.99, depending upon the range of distances
over which estimates were made. Similar indoor experiments yielded exponents
of 1.15 and 1.26. Obviously, unless these wide differences can be explained, it
is impossible to define perceived distance on the basis of these results. Even
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so, a definition of perceived distance is essential if one is to compare SDIH with
alternative hypotheses.
Interestingly, using the method of limits rather than magnitude estimation,
Cook (1978) obtained data that were well fitted by a power function with an
average exponent of approximately 1, albeit with large differences among subjects.
Sedgwick (2001) appears to accept this estimate. It is noteworthy that in Cook’s
study there was no attempt to control or monitor the criterion (response bias) of the
subjects. In any event, to a first approximation, his results suggest that if the classic
form of SDIH is correct, then physical distance may be used in place of perceived
distance to predict perceived size, thus reducing SDIH to Euclid’s law. However, the
perception of depth at long distances is less precise than it is over short distances,
if only because of the limited resolution of the human eye. Hence, we propose that
physical distance becomes an increasingly inaccurate measure of perceived distance
as distance increases.
The variability among magnitude estimation studies may well be due in part
to response bias, a topic addressed by detection theory, but normally not taken
into account in magnitude estimation experiments. In point of fact, introspective
reports concerning relative distances of objects that differ in perceived size are often
diametrically opposed to SDIH. Such reports are instances of the so-called size–
distance paradox (Gruber, 1954), which may well be attributable to a bias to decide
that the apparently larger of two objects is closer, even though other concurrently
present cues indicate otherwise.
The size–distance paradox led Sedgwick (1986) to express serious doubts about
SDIH. These doubts are shared by Ross and Plug (2002), among many others,
especially in connection with the moon illusion. For example, the apparent distance
theory of the moon illusion is based on the notion that the cues to distance afforded
by the terrain result in the horizon moon being perceived as more distant that the
zenith moon, which is viewed across an empty space. This ostensible difference
in perceived distance results in the horizon moon being perceived as larger than
the zenith moon, which is of equal angular size. However, when asked, subjects
generally report that the horizon moon is closer. Kaufman and Rock (1989) suggest
that this too may be an instance of a bias to decide that the object that appears to
be the larger is the closer object. Alternatively, the perceived size difference could
be a cue to relative distance, which happens to predominate despite the presence of
contrary cues. Several authors accept the latter possibility, and even deny that other
distance cues play a major role in producing differences in the perceived size of the
moon (e.g. Enright, 1989; McCready, 1986; Roscoe, 1989).
It is worth noting that the size–distance paradox is not the only reason for
questioning the applicability of SDIH to size perception. For example, Gibson
(1966, 1979) explicitly rejected the idea that distance cues are used by the
perceptual system in computing or inferring size. Assuming that the properties of
objects are uniquely represented in the optic array, Gibson (1966) explicitly states
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that there is no need to ascribe computation-like process to the brain in explaining
why we perceive what we do.
We begin with the proposition that the perceptual system takes distance into
account in achieving the perception of objective size. However, as stressed
previously, neither SDIH nor any alternative hypothesis can be evaluated without
a clear definition of psychological distance. We introduce the term perceptual
distance to stand for a criterion-free measure of psychological distance. The older
term perceived distance is avoided because of its ambiguous use in the history of
space perception.
Our definition relates perceptual distance to physical distance within the tradition
of Fechnerian psychophysics. To this end we conducted experiments based on
classic psychophysical methods, but modified to minimize effects of different
criteria.
EXPERIMENT 1
This experiment was designed to measure the uncertainty of depth discrimination
at different distances. To be clear, the term depth refers to the difference between
the distances to two objects, and may be thought of as an increment of distance.
If the average distance to two objects is kept constant, the depth between them
may be altered by moving one closer and the other farther away. The uncertainty of
detecting the depth between the objects depends upon the depth itself, and also upon
the average distance to the objects. As we shall see in the discussion, the variation
in this uncertainty with average distance may be used to compute how perceptual
distance varies with physical distance. In subsequent experiments we assess the
uncertainty of discriminating differences in objective size as a function of distance
under essentially the same conditions. This allows us to determine how perceived
size varies with perceptual distance.
The stimuli were luminous discs viewed across a virtual terrain that offered
information regarding the distance between the observer and the discs. The discs
were viewed stereoscopically, so that relative binocular disparity provided depth
information. It is important to emphasize that disparity alone reveals the order
of objects in depth, but cannot indicate the extent of the perceived depth. For
this disparity must be scaled by information (cues) regarding egocentric distance.
Hence, a virtual terrain was devised as a source of this information.
Method
Subjects. The same three subjects (LK, 77 year old male, RN, 59 year old male,
and SB 25 year old female) were employed in all of the experiments described in this
paper. All subjects had normal or corrected to normal vision and all had the same
interpupillary distance of approximately 6.6 cm. The participants were screened to
ensure that they were able to discriminate depth between stimuli based solely on
relative binocular disparity.
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Apparatus. To measure the uncertainty of depth discrimination as a function of
distance, subjects were presented with pairs of stimuli separated by relatively small
amounts of depth at each of four different distances. The probability of correctly
identifying which of the stimuli was closer (or more distant) was determined for
each value of depth.
All experiments were performed in a 10 × 20 laboratory in the Department of
Psychology on the C.W. Post Campus of Long Island University. The laboratory
walls were painted matte black. A fully silvered mirror, 8 × 4 , was mounted
vertically on one wall. A 6 × 4 partially silvered mirror was attached to a 4 × 8
vertical plywood partition parallel to and 9 feet (2.7 m) away from the fully silvered
mirror. The subject looked through a 2 × 2 window in the partition in front of the
partially silvered mirror and into the fully silvered mirror that faced it.
Subjects sat on an adjustable seat straight in front of a thin plate of glass,
which was at a 45◦ angle with respect to their line of sight (Fig. 1). A computer
monitor, to the left of the subject, displayed two moon-like discs measuring ∼0.5◦
in diameter. The distance between the centers of the discs on the monitor was 6.6 cm
(approximating the interpupillary distances of the subjects). A pair of 2 diopter
lenses, each 5 cm in diameter, and separated by 6.6 cm were placed parallel to and
50 cm away from the computer monitor. (The 50 cm focal lengths of the lenses
were verified by autocollimation.) Since the luminous discs on the display were
placed at the focus of each lens, their virtual images (seen as reflections in the thin
plate of glass) were at optical infinity. When separated by 6.6 cm, the absolute
parallax of the binocularly fused disc was zero deg, i.e. the lines of sight to the two
discs were parallel. Subjects were seated so that their eyes were at the same height
above the floor (∼117 cm) as were the centers of the two lenses. They fused the
discs simply by looking through the combining glass into the distance beyond the
partially silvered mirror.
A virtual terrain was created by strewing three hundred small white lamps in a
‘random’ array on the floor, which was covered by black felt, between the two
mirrors. To elevate the random pattern so that it was close to but still below the
moon-like stimuli, two black milk crates, one atop the other (total height = 75 cm),
were placed approximately half-way between the two large mirrors. The Christmas
tree lamps were draped across the topmost milk crate to produce an elevated terrain
lying just below the luminous disc stimulus. Figure 2 is a photograph of the scene
viewed by the subject.
The two parallel mirrors gave an impression similar to the repeated reflections
one might see when seated between two mirrors on the walls of a barbershop. The
mirror on the partition was partially silvered so the observer could sit outside the
9-foot space and view the multiple reflections of the terrain through the partially
silvered mirror in the more distant mirror on the wall. The lights were reflected
by the wall mirror back to the partially silvered mirror, and then back again, ad
infinitum. The first listed author was able to count about 17 reflections of the lamps
covering the crate located between the two mirrors. Hence, the visible extent of the
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Figure 1. Schematic of laboratory setup for size–distance experiments.
virtual terrain was about 46 meters in length. The density of the lamps increased
with distance, and the angular width of the terrain narrowed with distance.
The positions of the two discs on the monitor’s screen were under computer
control. Narrowing the separation of the two moons very slightly introduced a
small absolute binocular parallax [see Note 1]. In the experiments described here
the distances ranged from about 2.5 m to about 20 m, which translate to absolute
binocular parallaxes of ∼91 arc min to ∼11 arc min, respectively.
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Figure 2. A stereogram of the virtual terrain as seen by subjects. The images of each half-field were
taken with a hand-held digital camera (Canon Powershot G2, 35 mm FL equivalent). The camera
was placed to photograph the left eye’s view through the beam splitter, and then moved to the right to
photograph the stimulus array as seen by the right eye. The milk crate in the foreground was not as
distinctive as it is in the photos. Brightness and contrast were digitally adjusted to approximate how
the scene appeared to the subject. Fusing the two pictures with uncrossed eyes reveals a very distant
disc, which, in the actual scene, lay several dozen meters beyond the most distant lamps of the virtual
terrain.
A pixel 0.20 mm in width placed at the focus (50 cm) of the lens would permit
a relative disparity no smaller than 1.38 arc min [see Note 2]. To improve
this resolution, we took advantage of sub-pixel motion techniques. The relative
intensities of adjacent pixels were adjusted using 16 levels of grayscale, to obtain
translations of 1/16th pixel size. In principle this enables us to render translations
on the order of 6 arc sec, which would make it possible to present a stimulus at a
distance as great as 1.5 km before moving it to infinity (zero disparity). Thus, it
was possible to present small differences in binocular disparity within the range of
2 m to 1.5 km. As it turned out, in this first experiment the actual values of relative
disparity detected by our subjects was on the order of 1–3 arc min.
Within 2 meters of an observer all the usual cues of accommodation, convergence,
texture perspective, and binocular disparity are effective. The effectiveness of accommodation and convergence rapidly decline as distance increases. Accommodation is presumably ineffective beyond about 2 m (cf. Arditi, 1986), and convergence
(or, possibly, absolute binocular parallax per se) may be effective for distances as
great as 8 m (Foley, 1980). Relative binocular disparity and perspective are available well beyond 30 m. The apparent brightness of the terrain lights was visibly
graded with distance. Small lateral head movements allowed by this set up produced motion parallax in which the more distant lamps of the virtual terrain moved
with the observer, while those closer than the distance of fixation shifted in the
opposite direction.
Procedure. The stimuli were moon-like luminous discs, 0.5 deg in diameter. In
a typical trial a disc was presented at a particular distance. Subjects viewed this
disc as long as necessary to obtain an impression of its distance. Then subjects
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pressed a key and, 0.5 sec later (to minimize any apparent motion from one depth to
another), the disc was replaced by another identical disc that was either at the same
distance as the first, or at a slightly greater or lesser distance. The second disc was
placed ∼0.25 deg to the left of the first stimulus, so that the subject would remember
that he was to compare its distance to that of the preceding disc. Subjects then
had to decide if the second disc was closer or farther than the first, and signify the
decision by pressing key 1 or key 2 on the numeric keypad. Thus, the two-alternative
forced-choice (2AFC) procedure was employed. The effect of a bias to select the
first or second of two sequentially presented stimuli may be minimized in a 2AFC
experiment provided that the two alternatives are presented in random sequence and
have equal probabilities of occurrence. In this situation the subject’s criterion may
not vary over a large range. Hence, the proportion of correct responses could serve
as a relatively bias free index of discrimination (Green and Swets, 1966; Macmillian
and Creelman, 2005; Swets, 1996). This motivated our choice of the 2AFC method.
In any event, in all trials, either the first or second stimulus was at one of four
different ‘standard’ distances, i.e. 2.5, 5.0, 10.0 or 20.0 meters. The other stimulus
was at one of five preset ‘variable’ distances relative to each standard distance. One
of these was identical to the standard distance, two were more distant, and two less
distant. Hence, each of the four different ‘standard’ stimuli was paired with each
of five different ‘variable’ stimuli, making 20 possible pairs of stimuli. Since the
order of presentation of the ‘standard’ and ‘variable’ distances was interchanged,
there were 40 possible pairs of stimuli. These were presented in a random sequence
throughout the experiment until all pairs were presented 60 times, except for the
pairs when the standard and variable stimuli were identical, in which case the stimuli
were presented only 30 times. Therefore, 260 trials were conducted for each of the
four standard distances, making a grand total of 1040 trials for each subject in this
experiment.
The particular values of these ‘variable’ distances were established independently
for each of the three subjects. At first the subject saw a disc at one of the standard
distances, and the other disc was either greatly distant or much closer. The subjects
adjusted the disparity of this disc until it appeared to be at the same distance as
the disc at the standard distance. This procedure was repeated at least 20 times
by each subject and points of subjective equality (PSE) and estimated standard
deviations (σ ) of the responses were computed. The values of σ and 2 σ provided
first estimates of distances between the variable stimuli and each standard stimulus.
These estimates were further refined in pilot experiments employing the method
of constant stimuli to finally establish disparities appropriate to determining the
sensitivity of the subject to differences in disparity-determined depths at various
distances.
As intimated above, the 2AFC method was employed to minimize effects of
bias. It was combined with the method of constant stimuli to determine the per
cent correct responses for each depth surrounding each standard stimulus. These
data were fit to cumulative normal function by probit analysis. The PSE and σ
Size, depth and perceptual distance
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Figure 3. Log σdepth versus log PSEdist for three subjects. The abscissa represents the logarithm of
the PSEdist , i.e. the distances in cm at which the standard and variable discs are judged to be at the
same distance. The ordinate represents the logarithm of depth (in cm) equal to ±1 standard deviation
around the PSE. The linear functions indicate that σ is a power function of PSE.
of each resulting psychometric function estimated by this analysis are the data
contained in the Results section. The value of σ associated with each standard
distance represents the uncertainty of depth discrimination. It should be noted that
the reciprocal of σ may be defined as an index of the subject’s sensitivity to depth.
Results
The results are plotted as log σdepth versus log PSEdist in Fig. 3. The subscript ‘depth’
designates statistics based on performance of a depth discrimination task. PSE is
assigned the subscript ‘dist’ to signify that it is the point at which the variable and
standard discs are perceived as being at equal distances. Both depth and distance
are in units of centimeters. In general, σdepth increases monotonically with distance
for all participants. As shown in Fig. 3, log σdepth is a linear function of log PSEdist
for each of our three subjects. Hence, a power function describes the relationship
between σdepth and PSEdist . The slopes of these functions are 1.73 for LK, 1.79 for
SB, and 2.01 for RN. The fits to the straight lines are excellent, with r 2 = 0.99 for
LK, 0.98 for SB, and 0.99 for RN. Furthermore, the slopes of the three functions
do not differ significantly from each other (F = 1.15851, DF n = 2, DF d = 6,
p = 0.3754). Consequently, it is possible to calculate one slope for all the data.
The pooled slope equals 1.84, while the pooled Y intercept equals −3.57.
EXPERIMENT 2
We now turn to measuring the uncertainty of size discrimination. The angular size
of an object may remain constant as its distance is increased only if the object’s size
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is increased in proportion to its distance. If observers perceive objective size, then
they would see such an object grow in size with its distance.
A similar effect occurs with an after image, which may be likened to an object
of constant angular size. When it is projected onto a distant surface, the after
image appears to be larger than when it is projected onto a closer surface. Emmert’s
law holds that the perceived size of the after image is proportional to its perceived
distance, which is a restatement of SDIH. It will be recalled from Experiment 1 that
our 0.5 deg diameter discs had a constant size while being presented at different
distances. If distance information is used by the perceptual system in assessing
objective size, then how well we discriminate differences in size of a disc of constant
angular size as a function of its distance may clarify the nature of the connection
between size and distance perception. As will be made clear in the discussion, if the
uncertainty of depth discrimination parallels that associated with discriminating size
differences (associated with those depth differences), the two processes are likely to
be necessary to each other.
Method
Subjects. The three subjects of Experiment 1 were also employed in this
experiment.
Procedure. The 2AFC procedure was used in this experiment, but the subjects
were given different instructions than those of the Experiment 1. As before, a
moonlike disc was seen at one of the four standard ranges (2.5 m, 5 m, 10 m, and
20 m), and compared with another stimulus presented either at the same distance
or at a somewhat smaller or greater distance. Rather than decide which of the discs
was more distant, the subjects were instructed to determine which was the larger of
the two discs.
As in Experiment 1, the depth values assigned to each ‘variable’ distance in this
experiment were established by means of pilot trials, initially involving the method
of adjustment. The subjects simply moved a disc away from one of the four standard
positions until it appeared to differ in size. It quickly became obvious that subjects
were unable to detect differences in apparent size when the changes in depth were
approximately the same as those used in Experiment 1. The amount of depth had
to be greater. Further trials were conducted to establish depths that would make it
possible to employ the method of constant stimuli. These depths were larger than
those used in Experiment 1. Of course, this was an immediate indication that the
estimates of σ based on apparent size differences would be greater than obtained
when subjects were attempting to detect differences in depth.
In all of the pilot trials the disc’s angular size was constant (0.5 deg). However,
we recognized that if the angular sizes of the discs were always the same (as it was
in Experiment 1), subjects could use the perceptible difference in distance as a cue
to decide which was larger. In that case the results might be biased by a decision
that either the nearer or the more distant of the two discs was to be designated
Size, depth and perceptual distance
449
as the larger. To avoid this possibility, we added 180 trials at each of the four
standard ranges in which the two discs were of different angular sizes, i.e. the
0.5 deg diameter discs of the previous two experiments, and a second disc with an
angular diameter of 0.56 deg. As before, subjects were shown two discs on each trial
and had to decide which of the two was the larger. Hence, the task was not more
complicated than that of Experiment 1. Even so, on randomly selected trials the
two discs had the same angular diameters, either 0.56 or 0.5 deg, or had different
angular diameters. On half of those randomly interspersed trials the larger of the
two discs (0.56 deg) was the more distant, and, on the other half of the trials, the
angularly smaller of the discs was the more distant.
It is important to bear in mind that on any given trial the subject’s task was
essentially identical to that of Experiment 1. Confronted with a pair of discs, the
subject simply decided which of the two was the larger (or smaller) instead of which
was further away (or closer). Because of the inclusion of trials in which an angularly
larger disc was presented at a closer distance, the subject had no expectation that
a difference in size was associated with a particular difference in distance. The
angular sizes were chosen so that the angularly larger disc would be very likely to
be judged as larger when it was the closer, as well as when it was the more distant
member of the pair. This allowed the subjects to attend strictly to size, and ignore
differences in distance while making size judgments.
Owing to the use of two different sizes, the overall number of trials was increased
to 540 per subject at each of the five ranges. Thus, a total of 2700 trials were
obtained from each subject. The data were sorted so that results obtained when
the standard and variable discs were identical in size are presented separately from
those in which the discs differed in angular size.
Results
Figure 4 is based solely on those trials in which the two discs were equal in angular
size. It is similar to Fig. 3 in that the coordinates represent distance. The subscripts
reflect the task of the subject, which was to detect a difference in apparent size.
Thus, log PSEdist is the logarithm of the distance at which subjects were equally
likely to decide that the size of the variable disc was greater or less than that of the
standard disc. By the same token, the ordinate representing the logarithm of σsize
reflects the uncertainty in the amount of depth required to produce a difference in
perceived size.
The data points in Fig. 4 are well fitted by power functions where the exponents
were 1.71 for LK, 1.85 for SB, and 1.81 for RN. In all three cases the coefficient
of determination is 0.99, indicating that the fits to straight lines are excellent. This
is supported by a runs test, indicating that there is no significant departure from
linearity. Furthermore, the slopes of these three functions do not differ significantly
(F = 0.43268, DF n = 2, DF d = 6, p = 0.6675). The pooled slope equals
1.79. However, the Y intercepts differ very significantly (F = 69.9252, DF n = 2,
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Figure 4. Size judgments made by each subject. The abscissa represents the logarithm of the PSEdist ,
i.e. the distances at which the standard and variable discs are judged to be of the same size. The
ordinate is the logarithm of ±1 σ (also in cm) relative to the PSE.
Figure 5. Log σ for size and for depth versus log PSEdist with discs of equal angular size. The upper
line fits the pooled data of the three subjects of Experiment 2 (size uncertainty), while the lower line
fits the corresponding data of Experiment 1 (depth uncertainty). The difference in intercepts of the
two parallel functions is highly significant, with the elevation (Y intercept) of the function related to
size discrimination ∼0.8 log units higher than the function related to depth discrimination. The slopes
are essentially the same.
DF d = 6, p < 0.0001). Since the slopes are not significantly different, it is
possible to calculate one slope for all the data. The pooled slope equals 1.79.
Figure 5 summarizes the data of Figs 3 and 4. The plots are of lines that best fit
the pooled data across subjects in each of the two experiments.
The slopes of the two linear functions depicted in Fig. 5 do not differ significantly
(F = 0.197207, DF n = 1, DF d = 20, p = 0.6618). Since the slopes are
not significantly different, it is possible to calculate one slope for all the data,
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which equals 1.79. However, the elevations (Y intercepts) of the two functions
are extremely different from each other (F = 96.5397, DF n = 1, DF d = 20,
p < 0.0001), which signifies that uncertainty in size discrimination is much greater
than in depth discrimination. As is obvious in Fig. 5, this effect is present in all three
subjects, and is not due solely to the higher overall uncertainty of SB (see Fig. 4).
Finally, none of the subjects had any difficulty in deciding that the disc of larger
angular size was larger than the disc of smaller angular size. Thus, subject LK
identified the angularly larger disc correctly on 96% of all trials, regardless of
whether it was the nearer or more distant disc. Similarly, subject SB was correct
on 99% and RN on 94% of all trials. This result, together with the fact that in
both experiments subjects made a simple binary decision (i.e. one of the two discs
of each trial was the larger), supports the contention that the difference in overall
uncertainty in the size discrimination task is not attributable to the addition of the
extra discs of different angular size.
DISCUSSION
On perceptual distance
The reciprocal of σ is an index of sensitivity. Since σdepth is a power function of
distance with an exponent of ∼1.8, sensitivity to differences in distance varies as
1/D 1.8 . As anticipated, this represents a decaying ability to discriminate depth with
distance. In its original form, Fechner’s law is inapplicable to our data. However, it
is possible to re-scale distance so that σ is proportional to the transformed distance
(Falmagne, 1974, 1985; Kaufman, 1974; Luce and Galanter, 1963).
For any power law, σ (D ) = D α , this transformation is accomplished by taking
D → D 1/α , where D = perceptual distance. Hence, the slope of a psychometric
function describing the probability of discriminating a difference in the rescaled D would be the same for all D, i.e. the psychometric functions would all be parallel to
each other, thus resolving ‘Fechner’s problem’ (Luce and Galanter, 1963). Over the
range of our observations, a transformation relating perceptual distance to physical
distance raised to a power near 0.55 would be consistent with a Fechnerian law,
thereby providing an estimate of the relationship between perceptual and physical
distance. Hence, the law relating perceptual distance to physical distance has the
form of a power law, e.g.
log D ≈ 0.5 log D.
(2)
Such a law should not be confused with Steven’s law, which is predicated on
magnitude estimation rather than on the psychophysical (differential threshold)
methods basic to Fechnerian laws.
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Figure 6. Log σsize vs. log σdepth (all subjects). The slope of the function = 0.98 ± 0.13 and
r 2 = 0.85. The Y-intercept of ∼0.8 log units reflects the greater level of ‘noise’ associated with size
discrimination relative to depth discrimination.
Perceptual distance and size constancy
As is evident in Fig. 6, log σdepth (from Experiment 1) versus log σsize (from
Experiment 2) yields a linear function with a slope of ∼1. Thus, the uncertainty
associated with depth discrimination above the virtual terrain is proportional to the
uncertainty associated with size discrimination under the same condition. This
reinforces the connection between perceived size and perceptual distance. If
any transform were applied equally to both axes of Fig. 6 the result would be
a linear function having the same slope of ∼1.0. This proves that perceived
size is proportional to perceptual distance, thus confirming SDIH and, therefore,
supporting the proposition that size constancy may be achieved by taking perceptual
distance into account.
Perceptual distance and stereoscopic depth constancy
According to the geometry of stereopsis,
ρ = [α | D1 − D2 |]/[D1 × D2 ],
(3)
where ρ is the relative binocular disparity (in radians); α is the distance in meters
between the eyes; D1 is the distance in meters to one object; and D2 is the distance
in meters to a more distant object.
If we let (D2 − D1 ) = depth (δ), then
δ = [ρ(D1 × D2 )]/α.
(4)
Size, depth and perceptual distance
453
Basically, this equation states that the depth in meters between two objects is
proportional to the product D1 (D1 + δ) of their distances, which is equivalent to
ρ = αδ/D 2 ,
(5)
where D is the average of D1 and D2. It follows that
δ = (ρD 2 )/α.
(6)
Hence, if relative disparity is held constant, the magnitude of depth is approximately
proportional to D 2 .
Wallach and Zuckerman (1963) noted that this square law relationship predicts
depth constancy, just as SDIH predicts size constancy. They accepted the idea that in
size constancy the perceptual system compensates for perceived distance in accord
with Emmert’s law, which is essentially the same as SDIH. Depth constancy would
require that the system compensate for the square of the distance to the object.
Furthermore, they also presented convincing evidence that, over a limited range of
distances, in the presence of adequate cues to distance, the actual perceived depth
between two objects of constant disparity does indeed vary approximately with the
square of distance. As in size constancy, depth constancy is nearly perfect in a
full cue situation, and is seriously degraded in a minimal cue situation (Glennerster
et al., 1998).
Despite this similarity, Wallach and Zuckerman attributed size constancy and
stereoscopic depth constancy to different mechanisms. Although distance plays
a crucial role in both types of constancy, size constancy entails compensating
for distance per se, while depth constancy entails compensating for distance
raised approximately to the second power. Consequently, in their view achieving
constancy requires different processes for depth and for size.
The results of our experiments lead to a different conclusion. We propose
instead that wherever the cue of binocular disparity plays a role, precisely the same
mechanisms underlie perceived depth and perceived size.
The value of σdepth is a measure of uncertainty. But it may also be described as a
depth interval δ between two stimuli presented 0.5 sec apart in time. Both uncertainty and depth vary approximately as the square of distance, so our uncertainty
effect is essentially the same as stereoscopic depth constancy. As shown in Fig. 3
of Experiment 1, the slope of the pooled function relating σdepth to PSEdist was 1.8,
indicating that uncertainty increases approximately with the square of distance. As
stated in equation (6), given a constant relative binocular disparity, the geometry of
stereopsis requires that the depth between two objects be proportional to the square
of their average distance. When σ is pooled over subjects and expressed in terms of
relative binocular disparity rather than cm, and plotted against PSEdist , the slope of
the resulting empirical function is 0.01 arc min per cm. This slight departure from
a slope of zero indicates the presence of a very small departure from ideal depth
constancy in the direction of a slight underconstancy. Thus, depth constancy and
size constancy can be accounted for by the same approximation to a square law re-
454
L. Kaufman et al.
lationship between uncertainty and distance, providing the basis for a unified theory
of size and depth constancy. The change in depth sufficient to increase perceived
size varies with distance at a rate that is essentially the same as that producing a
perceptible change in depth (Fig. 6).
On the sequence of processing underlying size constancy
As illustrated by Fig. 5, plotting σ versus PSE, pooled over subjects for both tasks,
reveals a natural division of the data into two linear functions, one representing the
data of the depth discrimination task, and the other the size discrimination data. The
slopes of the two functions do not differ significantly, but their Y intercepts do. The
elevation of the size function is about 0.8 log units above that of the depth function.
This difference is potentially important in determining the sequence of processing
underlying objective size perception.
As mentioned in the Introduction, the assumption that distance is taken into
account in size perception begs a number of questions. One question, which we
have addressed, concerned defining the relationship between perceived and physical
distance. McKee and Welch (1992) addressed another question, namely, is angular
size processed prior to distance information, or are distance cues processed prior to
angular size?
To answer this question McKee and Welch used a method devised by Burbeck
(1987). They began with a model in which the precision of judging objective size
entails combining two independent processes, one being the neural measurement
of angular (retinal) size, and the other that of distance. They also assumed that
the discrimination of small differences in angular size is limited by noise. The
smallest detectable change in objective size is also limited by noise. The noise
involved in matching a particular angular size manifests itself in the variability of
such matches. A measure of this variability is its standard deviation (σ ), which,
as already stated, represents the uncertainty that limits the discrimination of a
difference. Similarly, the discrimination of a difference in objective size is also
limited by noise. If the noise associated with discriminating differences in objective
size is significantly greater than that associated with discriminating differences in
angular size, by implication, the encoding of angular size is prior to that of the
encoding of objective size. This follows from their model’s assumption that the
noise related to computing distance adds to the noise associated with computing
angular size when the perceiver is estimating objective size. Alternatively, angular
size may be inferred by discounting the effect of distance on the perceived objective
size. In that case, according to McKee and Welch’s model, the noise associated
with judging objective size would be less than that associated with judging angular
size. McKee and Welch tested these alternatives by comparing the uncertainty σ
associated with matches of objective size with corresponding values of σ obtained
in matches of angular size. As it turned out, their observers were unable to ignore
differences in depth when making angular size judgments, suggesting that their
observers did not have direct access to information about retinal (angular) size.
Size, depth and perceptual distance
455
Moreover, the results were inconclusive with regard to their alternative hypotheses.
McKee and Welch suggested that angular size and distance are processed in parallel
and neither depends directly on the other (see also McKee and Smallman, 1998).
However, since their subjects were unable to ignore distance in making angular
size judgments, this implies that comparing angular and objective size judgments
many not have been the appropriate measures to test the McKee and Welch
hypotheses. In fact, their results imply that angular size per se may not be available
to conscious perception, a conclusion already reached by Wallach and McKenna
(1960). It should be noted that others have disputed this conclusion (e.g. Rock and
McDermott, 1964),
Rather than comparing the noise associated with estimating angular size versus
the noise associated with estimating objective size, this paper compares the noise
associated with depth discrimination at different distances with that associated with
discriminating objective size differences produced by altering distance. Our results
lead us to conclude that the noise associated with objective size discrimination is
uniformly greater than that associated with depth discrimination. In the context
of the McKee and Welch model, this supports their hypothesis that distance is
processed prior to angular size.
Acknowledgements
This work was supported by NSF Grant No. BCS0137567, L. Kaufman, PI.
It was also supported in part by IBM Research. We are very grateful to JeanClaude Falmagne, Julian Hochberg, Zhong-Lin Lu, Ethel Matin, Arnold Trehub,
Hal Sedgwick, and Helen Ross who provided invaluable comments on an early
version of this paper. We are especially grateful to the editors and reviewers whose
many suggestions substantially improved this paper. Finally, we thank Vassias
Vassilliades for his assistance and creative ideas.
NOTES
1. Absolute parallax is synonymous with the term absolute disparity. The absolute
parallax of a point in space results from the fact that the two eyes view the same
point from two different positions. It is identical with the convergence angle needed
to get the images of the point centered in the two foveas (Kaufman, 1974).
2. The term relative disparity refers to the difference between the absolute
parallax of one point and that of another point at the same or some other distance.
Relative binocular disparity is the cue to binocular stereopsis.
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